Theorem dfrnf 4852

Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
dfrnf.1	⊢ Ⅎ𝑥𝐴
dfrnf.2	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
dfrnf	⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem dfrnf

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrn2 4799	. 2 ⊢ ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤}
2		nfcv 2312	. . . . 5 ⊢ Ⅎ𝑥𝑣
3		dfrnf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2312	. . . . 5 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfbr 4035	. . . 4 ⊢ Ⅎ𝑥 𝑣𝐴𝑤
6		nfv 1521	. . . 4 ⊢ Ⅎ𝑣 𝑥𝐴𝑤
7		breq1 3992	. . . 4 ⊢ (𝑣 = 𝑥 → (𝑣𝐴𝑤 ↔ 𝑥𝐴𝑤))
8	5, 6, 7	cbvex 1749	. . 3 ⊢ (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤)
9	8	abbii 2286	. 2 ⊢ {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤}
10		nfcv 2312	. . . . 5 ⊢ Ⅎ𝑦𝑥
11		dfrnf.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
12		nfcv 2312	. . . . 5 ⊢ Ⅎ𝑦𝑤
13	10, 11, 12	nfbr 4035	. . . 4 ⊢ Ⅎ𝑦 𝑥𝐴𝑤
14	13	nfex 1630	. . 3 ⊢ Ⅎ𝑦∃𝑥 𝑥𝐴𝑤
15		nfv 1521	. . 3 ⊢ Ⅎ𝑤∃𝑥 𝑥𝐴𝑦
16		breq2 3993	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑥𝐴𝑤 ↔ 𝑥𝐴𝑦))
17	16	exbidv 1818	. . 3 ⊢ (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦))
18	14, 15, 17	cbvab 2294	. 2 ⊢ {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
19	1, 9, 18	3eqtri 2195	1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Syntax hints:   = wceq 1348  ∃wex 1485  {cab 2156  Ⅎwnfc 2299   class class class wbr 3989  ran crn 4612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619  df-dm 4621  df-rn 4622

This theorem is referenced by:  rnopab  4858